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We consider the radial wave equation in similarity coordinates within thesemigroup formalism. It is known that the generator of the semigroup exhibits acontinuum of eigenvalues and embedded in this continuum there exists a discreteset of eigenvalues with analytic eigenfunctions. Our results show that, forsufficiently regular data, the long time behaviour of the solution is governedby the analytic eigenfunctions. The same techniques are applied to the linearstability problem for the fundamental self--similar solution $\chi_T$ of thewave equation with a focusing power nonlinearity. Analogous to the free waveequation, we show that the long time behaviour (in similarity coordinates) oflinear perturbations around $\chi_T$ is governed by analytic mode solutions. Inparticular, this yields a rigorous proof for the linear stability of $\chi_T$with the sharp decay rate for the perturbations.

Asymptotics and analytic modes for the wave equation in similarity coordinates